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Magazine Chapter 2: Problem 31
Sketch the graph of the circle or semicircle. $$4 x^{2}+4 y^{2}=1$$
Short Answer
Expert verified
The graph is a circle with center (0,0) and radius \(\frac{1}{2}\).
Step by step solution
01
Identify the Equation
The given equation is \(4x^2 + 4y^2 = 1\). This equation resembles the standard form of a circle equation, which is \((x-h)^2 + (y-k)^2 = r^2\), but with constant coefficients.
02
Simplify the Equation
Divide the whole equation by 4 to simplify it: \(x^2 + y^2 = \frac{1}{4}\). Now, the equation looks like \((x-0)^2 + (y-0)^2 = (\frac{1}{2})^2\).
03
Determine the Circle Properties
From the simplified equation \((x-0)^2 + (y-0)^2 = (\frac{1}{2})^2\), identify that the center of the circle is at (0,0) and the radius is \(\frac{1}{2}\).
04
Sketch the Circle
To sketch the circle, mark the center at the origin (0, 0) on a coordinate plane. Then, draw a circle with a radius of \(\frac{1}{2}\). The circle should be centered at the origin and will have a diameter of 1.
05
Verify the Graph
Ensure all points on your circle satisfy the equation \(x^2 + y^2 = \frac{1}{4}\). Pick a few points (for example, (0, \(\frac{1}{2}\)), (\(\frac{1}{2}\), 0), etc.) and verify they satisfy the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Equation of a Circle
A circle's equation is fundamentally derived from the Pythagorean Theorem and represents all the points that are equidistant from a certain point called the center. The general form of a circle's equation is
The combination of these squared terms sums to the square of the radius, \( r^2 \). Any point \( (x, y) \) on the circle maintains a constant distance from the center, which is defined by the value of \( r \).
- Centered at \( (h, k) \) with radius \( r \)
- Written as \( (x-h)^2 + (y-k)^2 = r^2 \)
The combination of these squared terms sums to the square of the radius, \( r^2 \). Any point \( (x, y) \) on the circle maintains a constant distance from the center, which is defined by the value of \( r \).
Standard Form of Circle Equation
The standard form of a circle's equation provides clarity on the circle's position and size on a coordinate plane. It is presented as
\( (x-0)^2 + (y-0)^2 = \(\( \frac{1}{2} \)\)^2 \)
illustrates a circle centered at the origin \( (0,0) \) with a radius of \( \frac{1}{2} \). This makes it easy to graph and interpret the circle's properties instantly from its equation.
- \((x-h)^2 + (y-k)^2 = r^2\)
- \((h, k)\) is the circle's center
- \(r\) is the circle's radius
\( (x-0)^2 + (y-0)^2 = \(\( \frac{1}{2} \)\)^2 \)
illustrates a circle centered at the origin \( (0,0) \) with a radius of \( \frac{1}{2} \). This makes it easy to graph and interpret the circle's properties instantly from its equation.
Circle Properties
Understanding the properties of a circle is vital to working with circles effectively. A circle has several key properties:
- **Center** - The fixed point from which all points on the circle are equidistant.
- **Radius** - The constant distance from the center to any point on the circle.
- **Diameter** - Twice the radius; the longest distance across the circle.
- Center at \( (0,0) \)
- Radius is \( \frac{1}{2} \)
- Diameter, therefore, is 1 (since \( 2 \times \frac{1}{2} = 1 \))
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph mathematical equations, including circles. It consists of:
- Two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
- A point of intersection between these axes, called the origin, marked as (0,0).
- Mark the circle's center on the plane according to its coordinates \( (h, k) \).
- Use the radius to draw a loop around the center, making sure all points on the circle maintain this fixed distance from the center.
- Verify that the plotted points satisfy the circle's equation. For example, in our exercise with a radius of \(\frac{1}{2}\), testing the points \( (0, \frac{1}{2}), (\frac{1}{2}, 0) \) ensures they lie on the circle.
